A193910 Leap centuries in the revised Julian calendar.

2, 6, 11, 15, 20, 24, 29, 33, 38, 42, 47, 51, 56, 60, 65, 69, 74, 78, 83, 87, 92, 96, 101, 105, 110, 114, 119, 123, 128, 132, 137, 141, 146, 150, 155, 159, 164, 168, 173, 177, 182, 186, 191, 195, 200, 204, 209, 213, 218, 222, 227, 231, 236, 240, 245, 249

Offset

1,1

Comments

Terms divided by 100, e.g., 29 indicates year 2900, which is a revised Julian leap year, but not a Gregorian leap year. Values below 20 are "proleptic" (only based on the formula).

Links

Table of n, a(n) for n=1..56.

M. Milankovitch, Das Ende des julianischen Kalenders und der neue Kalender der orientalischen Kirchen, Astronomische Nachrichten, volume 220 (1924), 379-384.

Claus Tøndering, Frequently Asked Questions about Calendars, Don't the Greeks do it differently?

Wikipedia, Revised Julian calendar

Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Formula

a(n) = a(n-2) + 9. - Charles R Greathouse IV, Aug 09 2011

a(n) = 2 or 6 (mod 9).

For all positive integers n, a(n) = (1/4)*(18*n-17*(-1)^n-11), which implies a(2*n-1) = 9*n-3 and a(2*n) = 9*n-7. - Farideh Firoozbakht, Oct 08 2014

G.f.: x*(2 + 4*x + 3*x^2)/((1 + x)*(1 - x)^2). - Philippe Deléham, Nov 30 2016

Example

20 mod 9 is 2; 2000 was a leap year in the revised Julian calendar.
24 mod 9 is 6; 2400 and 2000 also happen to be Gregorian leap years.
28 is the first integer greater than 16 only contained in A008586.
29 is the first integer greater than 16 not contained in A008586.

Mathematica

Table[1/4 (18 m - (-1)^m - 11), {m, 56}] (* Farideh Firoozbakht, Oct 08 2014 *)

Prog

(Rexx) do C = 0 to 250; J = C // 9; if J = 2 | J = 6 then say C; end C
(PARI) a(n)=(9*n-5)\2 \\ Charles R Greathouse IV, Aug 23 2011

Crossrefs

A008586 enumerates "Gregorian leap centuries" (N mod 4 = 0).

A193879 enumerates all differences from A008586.

Cf. A274406: numbers congruent to {0, 8} mod 9; A301451: numbers congruent to {1, 7} mod 9. This sequence lists the numbers congruent to {2, 6} mod 9.

Sequence in context: A299637 A190889 A020966 * A215918 A078476 A217687

Adjacent sequences: A193907 A193908 A193909 * A193911 A193912 A193913

Keyword

nonn,easy

Author

Frank Ellermann, Aug 09 2011

Status

approved